4 BANACH ALGEBRAS WITH INVOLUTION. STAR ALGEBRAS 343 17. Let A be a Banach algebra with unit element <?, and let x>->jt* be an involution on A which is not necessarily continuous. (a) Show that the radical SR of A (Section 1 5.2, Problem 7) is a self-adjoint subset of A. Passing to the quotient, it follows that the involution on A defines an involution (also written z*~*z*) on A/8ft. (b) Let a = a* be a self-adjoint element of A, such that SpA(#) is contained in the open half-plane &£ > 0. Then there exists an element b e A, which is the limit of a sequence (un) of self-adjoint elements which are polynomials in a, and which satisfies b2 = a. (Write a — oc(e — (e — a"^)) for a suitable a > 0, and use Problem ll(f) and Section 15.2, Problem 15.) Show that b and b* commute, and that we may there- fore write b = u -f iv, where u and v are self-adjoint and commute with each other and with a. Deduce that uv = 0 and u2 ~ v2 = a. (c) Let C be a commutative Banach subalgebra of A which contains e, a, b, b* and is such that Spc(&) = SpA(6) (Section 15.2, Problem 15). Let C' be a commutative Banach subalgebra of A containing the image of C under the homomorphism 9 : x*-*x* of C into A. If 9l7 is the radical of C' and vr : C' -> C'/ST the canonical homomorphism, then the homomorphism TT ° 99 : C-^C'/B' is continuous (Section 15.3, Problem 20). Deduce that v = — (b* — b) e SR', and thence that SpA(<a) = SpA(w2). (Observe that SpA(<2) = SpC'/$R/(7r(w2 — v2)) = Spc'/W'W"2)).) Consequently u is invertible, v = w~1wt? = 0, and finally & = w is self-adjoint. Furthermore, if Sp(a) is contained in ]0, 4- oo [, then the same is true of Sp(6). 18. With the hypotheses of Problem 17, put pA(x) = (p(je*x))1/2 (also denoted by p(x)). We have X**) = P(X) ; also if s$ is the radical of A and if TT : A -> A/sJt is the canonical homomorphism, then ^A/wC^W) = PA(X). The involution jci-^x* on A is said to be hermitian, and the involutive algebra A is said to be hermit Ian, if Sp(a) c R for all self-adjoint elements a E A. If A is com- mutative, an equivalent condition is that every character of A is hermitian (Problem 2). Assume for the rest of this Problem that A is hermitian. (a) Let x e A be such that p(x) < I . Show that (e -f x*)(e — x) is invertible. (Remark that e — x*x — w2, where w is self-adjoint and invertible, by virtue of Problem 17; hence show that the spectrum of w~l(x* — x)w"x is contained in z'R.) Deduce that for all x e A (Ptdk's inequality), Consequently, if x is normal, we have p(x) —p(x). (Remark that if x, y are two elements of a Banach algebra which commute, then p(xy) ^p(x)p(y), by (15.3.4).) In particular, if x is unitary, then p(x) — I and Sp(x) c U. (b) Deduce from (a) that for each pair a, b of self-adjoint elements of A we have p(ab) <; p(a)p(b). (Observe that p(ab) <>p(ab) « (p(a2b2))lf2 <; ||a2||1/2 • I|62II1/2.) For all x, y e A we have p(xy) < p(x)p(y), and the radical of A is the set of all x e A such that p(x) = 0. (c) Suppose that a, b are self-adjoint elements of A such that Sp(<a) and Sp(6) are contained in [0, -f oo [. Show that Sp(a -f- b) is contained in [0, + oo [. (Write with p(u) < I and p(v) < 1 .) Deduce that for any two self-adjoint elements cz, b of A, we have p(a + b) <£ p(a) + p(b). Show also that Sp(;t*je + xx*) <= [0, + co [ for all x e A. the real vector subspace H of A con-